Homework Help:
Set Theory, One to One Correspondence

Show that the equation f(m,n) = 2^m(2n+1)-1 defines a one-to-one correspondence between w x w to w.

Where w (omega) is a symbol used to represent the 0,1,2,3,4,5,6...

Question: The book defines a one to one correspondence as a one to one function from A onto B. Is this in effect a bijection since its also onto?

Further Question: The next question asks to find a one to one correspondence between [0,1] and (0,1) in the reals. This leads me to believe that the function does not have to be onto since I dont believe it is possible to have a bijection between a closed dense set to an open one. What do you make of this?

3. The attempt at a solution

I keep coming back to induction though it gets messy quickly and I cant seem to resolve some of my problems.

Try to describe a method to find the inverse. If I give you N, how would you try to find an m and n such that N=f(m,n)? Hint: think about the prime factorization of N+1. Once you get that, you may be able to answer the 'onto' question as well.

Right. That is a perfectly rigorous proof. You've given a perfectly deterministic and unique way of finding m and n given N. And, sure, it depends on unique prime factorization. But I don't think you can do it any other way.